چکیده انگلیسی

The safety-first principle is a natural motivational factor in decision making, and is closely related to certain popular heuristics such as satisficing. We provide a systematic analysis of optimal portfolio choice under Roy’s safety-first principle by examining and comparing the behavior patterns of three popular investment strategies: the optimal constant-rebalanced portfolio, dynamic-rebalanced portfolio and buy-and-hold strategies. Our results indicate the importance of a match between the investment strategy, the investment goal, and the investment horizon. We also develop a geometric approach to investigate the relationships among the safety-first, expected utility, and mean-variance models and offer an explanation for the long-standing debate concerning different patterns of time-diversification effects.

مقدمه انگلیسی

Establishing appropriate performance criteria for investment decisions and deriving the corresponding best portfolio policies are two interrelated and fundamental tasks that must be accomplished when investing in financial markets. As indicated in Markowitz (1999) and widely accepted by the research community, the development of modern portfolio theory has its origin in two seminal papers published in 1952: Markowitz (1952) and Roy (1952). In the first pioneering work, Markowitz (1952) develops the well-known mean-variance (MV) approach under which investors have to achieve a balance between two conflicting objectives: maximizing the expected return of a portfolio and minimizing the investment risk measured by the portfolio variance. In the second pioneering work, Roy (1952) proposes the so-called safety-first principle, which suggests that investors behave in such a way that the probability of the portfolio value falling below a specified disaster level is minimized,1 i.e., the investment objective under Roy’s safety-first principle is to minimize the ruin probability or maximize the chance of survival. Mathematically, the safety-first principle can be formulated in the following way
As the natural choice in many situations, Roy’s safety-first approach laid the foundation for many later developments in finance, most prominently those in behavioral finance and risk management. In a certain sense, the subsistence level in the safety-first principle can also be viewed as the satisfaction level. Simon (1955), who introduces the term bounded rationality (also called limited rationality), suggests that individuals should simplify their decision problems using the satisficing approach: satisfactory if the target is reached or unsatisfactory if it is not. Hence, Roy’s safety-first principle can also serve as a behavioral criterion for rational choices. A notable application of this principle in behavioral finance can be found in the work of Shefrin and Statman (2000). The most widely adopted risk measure in risk management, Value-at-Risk (VaR) (see, e.g., Duffie and Pan, 1997 and Jorion, 1997), and many other downside risk measures are rooted in Roy’s safety-first principle (see, e.g., Bawa, 1976, Bawa, 1978, Basak and Shapiro, 2001 and Nawrocki, 1999).
The past decade or so has witnessed a number of studies related to this safety-first principle. For example, Roy (1995) develops a discrete-time dynamic optimization problem in which the objective is to maximize the long-run probability of survival. Li, Chan and Ng (1998) solve a multi-period safety-first portfolio selection problem by converting it into a corresponding mean-variance formulation using the Chebychev inequality. Adopting a continuous-time market setting, the papers by Browne, 1995, Browne, 1997, Browne, 1999a, Browne, 1999b and Browne, 1999c derive the optimal dynamic investment strategy for a portfolio manager who seeks to maximize the probability of reaching a particular goal.2Föllmer and Leukert (1999) introduce the safety-first concept into dynamic hedging problems. The papers by Stutzer, 2000 and Stutzer, 2003 draw on this concept to propose a new portfolio selection criterion that maximizes the decay rate of the probability of realizing a portfolio return below some predetermined target return level. Chiu and Li (2009) extend Roy’s safety-first principle to the asset-liability management problem. It should be noted that most of the above mentioned papers deal with infinite horizon problems, rather than those with a finite time horizon. Furthermore, compared to the study of dynamic mean-variance portfolio selection (see, for example, Li and Ng, 2000, Zhou and Li, 2000, Lim and Zhou, 2002, Zhou and Yin, 2003, Yin and Zhou, 2004 and Bielecki et al., 2005), a systematic analysis of different types of investment policies under the safety-first principle remains lacking.
We thus devote this paper to a systematic investigation of investment decision behavior patterns under Roy’s safety-first principle. To provide a comprehensive analysis of this principle, this paper examines and compares the three most popular portfolio selection strategies adopted in both academic study and real investment practice (see, e.g., Brandt, 1999 and Barberis, 2000): the buy-and-hold, the dynamic-rebalanced portfolio (DRP) and constant-rebalanced portfolio (CRP) strategies. In a buy-and-hold strategy, the determination of a portfolio occurs only at the beginning of the investment horizon. A DRP strategy allows investors with maximum freedom to dynamically change the weights of the individual securities in the portfolio over time. A CRP strategy, in contrast, retains the same distribution of wealth among the securities all of the time; that is, the proportion of the investor’s total wealth that is invested in each of the underlying securities remains the same all of the time. It should be noted that such a CRP strategy still implies continuous-trading, as one has to trade at every time instant to ensure that the investment proportions are rebalanced back to their original settings as the stock prices move up and down. A variety of optimality properties have been found for the CRP policies in ordinary portfolio selection, and these have been widely used in asset allocation practice (see, e.g., Perold & Sharpe, 1988).
In this paper, we examine how different types of investment strategies work under the safety-first principle. Within our framework, there are three dimensions for the portfolio choice problem: the investment strategy, the investment horizon and the investment goal. By comparing the optimal solutions of the different strategies, we intend to address an issue that in which situation a certain investment strategy confers a relative advantage.
We also establish the relationships between the time-diversification effect (i.e., the horizon effect) and the investment’s target growth rate based on the optimal portfolio choice under the safety-first principle. Different patterns exist for the horizon effect in the financial literature. In spite of the vast amount of work carried out on the topic of the horizon effect, controversy remains about its implications for optimal asset allocations. Prevalent in the investment world is a belief in the time-diversification effect: the longer the investment horizon, the more that an investor should invest in stocks. However a number of empirical studies support the existence of a reverse time-diversification effect. For example, Ameriks and Zeldes (2001) report that, ceteris paribus, elderly people typically hold more risky positions than do younger people. However, these opposite effects cannot be justified by risk aversion in the traditional expected utility framework. Merton, 1969 and Merton, 1971 and Samuelson (1969) both find that if asset returns are of the nature of a random walk, then the optimal strategy for an investor with a Constant Relative Risk Aversion (CRRA) utility is independent of the investment horizon, i.e., there is no horizon effect in determining the asset allocation. In this paper, we demonstrate that different horizon effect patterns can be intuitively understood within the safety-first framework. More specifically, our results show that the investment decisions generated under the safety-first principle with different subsistence level patterns demonstrate different horizon effect patterns. For example, a subsistence level that grows at a faster rate than the exponential rate yields the time-diversification effect. By considering both the investment goal and the investment horizon under the safety-first principle, we offer an explanation for the long-standing debate concerning the different time-diversification effects.
The contributions of the research presented in this paper can be summarized as follows.
•We provide a rather detailed examination of the safety-first portfolio selection problem from the perspective of three basic trading strategies: the buy-and-hold, DRP and CRP strategies.
•We investigate the relative advantages and disadvantages of active and passive trading behaviors under the safety-first principle.
•We develop a geometric approach to investigate the relationships among the safety-first, expected utility and mean-variance models under a continuous-time setting.
•We establish the relationship between the time-diversification effects and an investment’s target growth rate within the framework of the safety-first principle.
The reminder of the paper is organized as follows. In Section 2, we introduce our problem setting under the safety-first principle in a Black–Scholes continuous-time market. In Section 3, we summarize the closed-form expressions for the three types of optimal strategies under the safety-first formulation. We extend the results assuming a constant subsistence level to a problem with a dynamic subsistence level in Section 4. In Section 5, we contrast the three types of optimal strategies by comparing their terminal wealth distributions under different performance measures, thus revealing some of the rich properties of the safety-first principle. Motivated by the research presented in Pyle and Turnovsky, 1970 and Pyle and Turnovsky, 1971 in a one-period context, we devote Section 6 to the development of geometric analysis for the optimal CRP strategy under the safety-first principle, expected utility and the mean-variance models. We then proceed in Section 6 to show the different patterns of horizon effects under the safety-first principle. We conclude our paper in Section 7.

نتیجه گیری انگلیسی

Although the expected utility or the use of coherent risk measures can provide rational investors with better investment guidance, Roy’s safety-first principle, which minimizes the probability of falling below a satisfaction level or, equivalently, maximizes the probability of reaching a given goal, remains a natural, and thus popular, choice for many investors and serves as a fundamental decision-making philosophy (e.g., Domian, Louton, & Racine, 2007). By investigating the three most popular portfolio selection strategies, the buy-and-hold, dynamic-rebalanced portfolio, and constant-rebalanced portfolio strategies, under Roy’s safety-first principle in a Black–Scholes’ market, we have not only presented the analytical properties of the safety-first principle in a more elegant and clearer way, relative to the previous literature, but also, and more importantly, we have explored a number of new features underlying this principle, thus furthering our understanding of it. Some of our notable results concerning this investment principle can be summarized as follows.
(i)An investor who follows the safety-first principle is target-first, rather than safety-first. More specifically, the degree of risk-taking in the safety-first policy largely depends on the difference between the present value of the satisfaction level and the initial wealth. In general, the safety-first principle does not necessarily lead to a conservative investment policy.
(ii)Although the optimal dynamic-rebalanced portfolio strategy under the safety-first principle minimizes the probability below the satisfaction level, it also involves investors not only in a loss of significant magnitude, but also causes them to give up the potential high returns from a stock.
(iii)Caution should be exercised in matching the investment strategy, investment horizon, and investment goal. The dynamic-rebalanced portfolio strategy performs best in terms of the final wealth only when the satisfaction level is set high with respect to the investment time horizon. In contrast, passive investment strategies, such as the constant-rebalanced and buy-and-hold strategies, become more suitable when the investment target is not very high compared to the investment horizon.
(iv)Under the safety-first principle, a growing subsistence level with a rate faster than the exponential rate implies a time-diversification effect, whereas a growing subsistence level with a rate slower than the exponential rate implies a reverse time-diversification effect.
(v)An investor’s degree of risk aversion can be estimated from the quality of the assets in which he or she invests (as measured by the market price of risk) and the wealth growth rate he or she would like to achieve.
We have also offered in this paper clear, diagrammatic representations of the optimal portfolio choices for investors under the safety-first principle or the CRRA utility by utilizing the continuous-time mean-variance framework. The geometric approach derived in selecting the optimal CRP strategy of the safety-first model and the expected utility model is clear and intuitive. Its extension can also be used in the study of many other portfolio selection problems, such as other forms of safety-first rules (e.g., Telser, 1956 and Kataoka, 1963).
Finally, we have to emphasize that the new insights on investment obtained in this paper are derived from a narrow context of the continuous-time safety-first formulation. It will be interesting to investigate whether these conclusions hold for some extended versions of the safety-first formulation (e.g., the Telser’s (1956) safety-first formulation) or in some other less restrictive model settings. Note that, while the ruin probability is the only performance measure in Roy’s safety-first formulation, most mean-risk portfolio selection formulations derive their investment decisions by balancing the expected wealth with a risk measure, for example, the Telser’s safety-first formulation (Telser, 1956) reaches its investment decision